Advance Analytics with R (UG 21-24)
I am Ayush.
I am a researcher working at the intersection of data, law, development and economics.
I teach Data Science using R at Gokhale Institute of Politics and Economics
I am a RStudio (Posit) certified tidyverse Instructor.
I am a Researcher at Oxford Poverty and Human development Initiative (OPHI), at the University of Oxford.
Reach me
ayush.ap58@gmail.com
ayush.patel@gipe.ac.in
Learn to apply and interpret simple and multiple linear regression models.
References for this lecture:
| ...1 | TV | radio | newspaper | sales |
|---|---|---|---|---|
| 1 | 230.1 | 37.8 | 69.2 | 22.1 |
| 2 | 44.5 | 39.3 | 45.1 | 10.4 |
| 3 | 17.2 | 45.9 | 69.3 | 9.3 |
| 4 | 151.5 | 41.3 | 58.5 | 18.5 |
| 5 | 180.8 | 10.8 | 58.4 | 12.9 |
| 6 | 8.7 | 48.9 | 75.0 | 7.2 |
| 7 | 57.5 | 32.8 | 23.5 | 11.8 |
| 8 | 120.2 | 19.6 | 11.6 | 13.2 |
| 9 | 8.6 | 2.1 | 1.0 | 4.8 |
| 10 | 199.8 | 2.6 | 21.2 | 10.6 |
| 11 | 66.1 | 5.8 | 24.2 | 8.6 |
| 12 | 214.7 | 24.0 | 4.0 | 17.4 |
| 13 | 23.8 | 35.1 | 65.9 | 9.2 |
| 14 | 97.5 | 7.6 | 7.2 | 9.7 |
| 15 | 204.1 | 32.9 | 46.0 | 19.0 |
| 16 | 195.4 | 47.7 | 52.9 | 22.4 |
| 17 | 67.8 | 36.6 | 114.0 | 12.5 |
| 18 | 281.4 | 39.6 | 55.8 | 24.4 |
| 19 | 69.2 | 20.5 | 18.3 | 11.3 |
| 20 | 147.3 | 23.9 | 19.1 | 14.6 |
| 21 | 218.4 | 27.7 | 53.4 | 18.0 |
| 22 | 237.4 | 5.1 | 23.5 | 12.5 |
| 23 | 13.2 | 15.9 | 49.6 | 5.6 |
| 24 | 228.3 | 16.9 | 26.2 | 15.5 |
| 25 | 62.3 | 12.6 | 18.3 | 9.7 |
| 26 | 262.9 | 3.5 | 19.5 | 12.0 |
| 27 | 142.9 | 29.3 | 12.6 | 15.0 |
| 28 | 240.1 | 16.7 | 22.9 | 15.9 |
| 29 | 248.8 | 27.1 | 22.9 | 18.9 |
| 30 | 70.6 | 16.0 | 40.8 | 10.5 |
| 31 | 292.9 | 28.3 | 43.2 | 21.4 |
| 32 | 112.9 | 17.4 | 38.6 | 11.9 |
| 33 | 97.2 | 1.5 | 30.0 | 9.6 |
| 34 | 265.6 | 20.0 | 0.3 | 17.4 |
| 35 | 95.7 | 1.4 | 7.4 | 9.5 |
| 36 | 290.7 | 4.1 | 8.5 | 12.8 |
| 37 | 266.9 | 43.8 | 5.0 | 25.4 |
| 38 | 74.7 | 49.4 | 45.7 | 14.7 |
| 39 | 43.1 | 26.7 | 35.1 | 10.1 |
| 40 | 228.0 | 37.7 | 32.0 | 21.5 |
| 41 | 202.5 | 22.3 | 31.6 | 16.6 |
| 42 | 177.0 | 33.4 | 38.7 | 17.1 |
| 43 | 293.6 | 27.7 | 1.8 | 20.7 |
| 44 | 206.9 | 8.4 | 26.4 | 12.9 |
| 45 | 25.1 | 25.7 | 43.3 | 8.5 |
| 46 | 175.1 | 22.5 | 31.5 | 14.9 |
| 47 | 89.7 | 9.9 | 35.7 | 10.6 |
| 48 | 239.9 | 41.5 | 18.5 | 23.2 |
| 49 | 227.2 | 15.8 | 49.9 | 14.8 |
| 50 | 66.9 | 11.7 | 36.8 | 9.7 |
| 51 | 199.8 | 3.1 | 34.6 | 11.4 |
| 52 | 100.4 | 9.6 | 3.6 | 10.7 |
| 53 | 216.4 | 41.7 | 39.6 | 22.6 |
| 54 | 182.6 | 46.2 | 58.7 | 21.2 |
| 55 | 262.7 | 28.8 | 15.9 | 20.2 |
| 56 | 198.9 | 49.4 | 60.0 | 23.7 |
| 57 | 7.3 | 28.1 | 41.4 | 5.5 |
| 58 | 136.2 | 19.2 | 16.6 | 13.2 |
| 59 | 210.8 | 49.6 | 37.7 | 23.8 |
| 60 | 210.7 | 29.5 | 9.3 | 18.4 |
| 61 | 53.5 | 2.0 | 21.4 | 8.1 |
| 62 | 261.3 | 42.7 | 54.7 | 24.2 |
| 63 | 239.3 | 15.5 | 27.3 | 15.7 |
| 64 | 102.7 | 29.6 | 8.4 | 14.0 |
| 65 | 131.1 | 42.8 | 28.9 | 18.0 |
| 66 | 69.0 | 9.3 | 0.9 | 9.3 |
| 67 | 31.5 | 24.6 | 2.2 | 9.5 |
| 68 | 139.3 | 14.5 | 10.2 | 13.4 |
| 69 | 237.4 | 27.5 | 11.0 | 18.9 |
| 70 | 216.8 | 43.9 | 27.2 | 22.3 |
| 71 | 199.1 | 30.6 | 38.7 | 18.3 |
| 72 | 109.8 | 14.3 | 31.7 | 12.4 |
| 73 | 26.8 | 33.0 | 19.3 | 8.8 |
| 74 | 129.4 | 5.7 | 31.3 | 11.0 |
| 75 | 213.4 | 24.6 | 13.1 | 17.0 |
| 76 | 16.9 | 43.7 | 89.4 | 8.7 |
| 77 | 27.5 | 1.6 | 20.7 | 6.9 |
| 78 | 120.5 | 28.5 | 14.2 | 14.2 |
| 79 | 5.4 | 29.9 | 9.4 | 5.3 |
| 80 | 116.0 | 7.7 | 23.1 | 11.0 |
| 81 | 76.4 | 26.7 | 22.3 | 11.8 |
| 82 | 239.8 | 4.1 | 36.9 | 12.3 |
| 83 | 75.3 | 20.3 | 32.5 | 11.3 |
| 84 | 68.4 | 44.5 | 35.6 | 13.6 |
| 85 | 213.5 | 43.0 | 33.8 | 21.7 |
| 86 | 193.2 | 18.4 | 65.7 | 15.2 |
| 87 | 76.3 | 27.5 | 16.0 | 12.0 |
| 88 | 110.7 | 40.6 | 63.2 | 16.0 |
| 89 | 88.3 | 25.5 | 73.4 | 12.9 |
| 90 | 109.8 | 47.8 | 51.4 | 16.7 |
| 91 | 134.3 | 4.9 | 9.3 | 11.2 |
| 92 | 28.6 | 1.5 | 33.0 | 7.3 |
| 93 | 217.7 | 33.5 | 59.0 | 19.4 |
| 94 | 250.9 | 36.5 | 72.3 | 22.2 |
| 95 | 107.4 | 14.0 | 10.9 | 11.5 |
| 96 | 163.3 | 31.6 | 52.9 | 16.9 |
| 97 | 197.6 | 3.5 | 5.9 | 11.7 |
| 98 | 184.9 | 21.0 | 22.0 | 15.5 |
| 99 | 289.7 | 42.3 | 51.2 | 25.4 |
| 100 | 135.2 | 41.7 | 45.9 | 17.2 |
| 101 | 222.4 | 4.3 | 49.8 | 11.7 |
| 102 | 296.4 | 36.3 | 100.9 | 23.8 |
| 103 | 280.2 | 10.1 | 21.4 | 14.8 |
| 104 | 187.9 | 17.2 | 17.9 | 14.7 |
| 105 | 238.2 | 34.3 | 5.3 | 20.7 |
| 106 | 137.9 | 46.4 | 59.0 | 19.2 |
| 107 | 25.0 | 11.0 | 29.7 | 7.2 |
| 108 | 90.4 | 0.3 | 23.2 | 8.7 |
| 109 | 13.1 | 0.4 | 25.6 | 5.3 |
| 110 | 255.4 | 26.9 | 5.5 | 19.8 |
| 111 | 225.8 | 8.2 | 56.5 | 13.4 |
| 112 | 241.7 | 38.0 | 23.2 | 21.8 |
| 113 | 175.7 | 15.4 | 2.4 | 14.1 |
| 114 | 209.6 | 20.6 | 10.7 | 15.9 |
| 115 | 78.2 | 46.8 | 34.5 | 14.6 |
| 116 | 75.1 | 35.0 | 52.7 | 12.6 |
| 117 | 139.2 | 14.3 | 25.6 | 12.2 |
| 118 | 76.4 | 0.8 | 14.8 | 9.4 |
| 119 | 125.7 | 36.9 | 79.2 | 15.9 |
| 120 | 19.4 | 16.0 | 22.3 | 6.6 |
| 121 | 141.3 | 26.8 | 46.2 | 15.5 |
| 122 | 18.8 | 21.7 | 50.4 | 7.0 |
| 123 | 224.0 | 2.4 | 15.6 | 11.6 |
| 124 | 123.1 | 34.6 | 12.4 | 15.2 |
| 125 | 229.5 | 32.3 | 74.2 | 19.7 |
| 126 | 87.2 | 11.8 | 25.9 | 10.6 |
| 127 | 7.8 | 38.9 | 50.6 | 6.6 |
| 128 | 80.2 | 0.0 | 9.2 | 8.8 |
| 129 | 220.3 | 49.0 | 3.2 | 24.7 |
| 130 | 59.6 | 12.0 | 43.1 | 9.7 |
| 131 | 0.7 | 39.6 | 8.7 | 1.6 |
| 132 | 265.2 | 2.9 | 43.0 | 12.7 |
| 133 | 8.4 | 27.2 | 2.1 | 5.7 |
| 134 | 219.8 | 33.5 | 45.1 | 19.6 |
| 135 | 36.9 | 38.6 | 65.6 | 10.8 |
| 136 | 48.3 | 47.0 | 8.5 | 11.6 |
| 137 | 25.6 | 39.0 | 9.3 | 9.5 |
| 138 | 273.7 | 28.9 | 59.7 | 20.8 |
| 139 | 43.0 | 25.9 | 20.5 | 9.6 |
| 140 | 184.9 | 43.9 | 1.7 | 20.7 |
| 141 | 73.4 | 17.0 | 12.9 | 10.9 |
| 142 | 193.7 | 35.4 | 75.6 | 19.2 |
| 143 | 220.5 | 33.2 | 37.9 | 20.1 |
| 144 | 104.6 | 5.7 | 34.4 | 10.4 |
| 145 | 96.2 | 14.8 | 38.9 | 11.4 |
| 146 | 140.3 | 1.9 | 9.0 | 10.3 |
| 147 | 240.1 | 7.3 | 8.7 | 13.2 |
| 148 | 243.2 | 49.0 | 44.3 | 25.4 |
| 149 | 38.0 | 40.3 | 11.9 | 10.9 |
| 150 | 44.7 | 25.8 | 20.6 | 10.1 |
| 151 | 280.7 | 13.9 | 37.0 | 16.1 |
| 152 | 121.0 | 8.4 | 48.7 | 11.6 |
| 153 | 197.6 | 23.3 | 14.2 | 16.6 |
| 154 | 171.3 | 39.7 | 37.7 | 19.0 |
| 155 | 187.8 | 21.1 | 9.5 | 15.6 |
| 156 | 4.1 | 11.6 | 5.7 | 3.2 |
| 157 | 93.9 | 43.5 | 50.5 | 15.3 |
| 158 | 149.8 | 1.3 | 24.3 | 10.1 |
| 159 | 11.7 | 36.9 | 45.2 | 7.3 |
| 160 | 131.7 | 18.4 | 34.6 | 12.9 |
| 161 | 172.5 | 18.1 | 30.7 | 14.4 |
| 162 | 85.7 | 35.8 | 49.3 | 13.3 |
| 163 | 188.4 | 18.1 | 25.6 | 14.9 |
| 164 | 163.5 | 36.8 | 7.4 | 18.0 |
| 165 | 117.2 | 14.7 | 5.4 | 11.9 |
| 166 | 234.5 | 3.4 | 84.8 | 11.9 |
| 167 | 17.9 | 37.6 | 21.6 | 8.0 |
| 168 | 206.8 | 5.2 | 19.4 | 12.2 |
| 169 | 215.4 | 23.6 | 57.6 | 17.1 |
| 170 | 284.3 | 10.6 | 6.4 | 15.0 |
| 171 | 50.0 | 11.6 | 18.4 | 8.4 |
| 172 | 164.5 | 20.9 | 47.4 | 14.5 |
| 173 | 19.6 | 20.1 | 17.0 | 7.6 |
| 174 | 168.4 | 7.1 | 12.8 | 11.7 |
| 175 | 222.4 | 3.4 | 13.1 | 11.5 |
| 176 | 276.9 | 48.9 | 41.8 | 27.0 |
| 177 | 248.4 | 30.2 | 20.3 | 20.2 |
| 178 | 170.2 | 7.8 | 35.2 | 11.7 |
| 179 | 276.7 | 2.3 | 23.7 | 11.8 |
| 180 | 165.6 | 10.0 | 17.6 | 12.6 |
| 181 | 156.6 | 2.6 | 8.3 | 10.5 |
| 182 | 218.5 | 5.4 | 27.4 | 12.2 |
| 183 | 56.2 | 5.7 | 29.7 | 8.7 |
| 184 | 287.6 | 43.0 | 71.8 | 26.2 |
| 185 | 253.8 | 21.3 | 30.0 | 17.6 |
| 186 | 205.0 | 45.1 | 19.6 | 22.6 |
| 187 | 139.5 | 2.1 | 26.6 | 10.3 |
| 188 | 191.1 | 28.7 | 18.2 | 17.3 |
| 189 | 286.0 | 13.9 | 3.7 | 15.9 |
| 190 | 18.7 | 12.1 | 23.4 | 6.7 |
| 191 | 39.5 | 41.1 | 5.8 | 10.8 |
| 192 | 75.5 | 10.8 | 6.0 | 9.9 |
| 193 | 17.2 | 4.1 | 31.6 | 5.9 |
| 194 | 166.8 | 42.0 | 3.6 | 19.6 |
| 195 | 149.7 | 35.6 | 6.0 | 17.3 |
| 196 | 38.2 | 3.7 | 13.8 | 7.6 |
| 197 | 94.2 | 4.9 | 8.1 | 9.7 |
| 198 | 177.0 | 9.3 | 6.4 | 12.8 |
| 199 | 283.6 | 42.0 | 66.2 | 25.5 |
| 200 | 232.1 | 8.6 | 8.7 | 13.4 |
sales and TV
[1] 0.7822244
sales and radio
[1] 0.5762226
sales and newspaper
[1] 0.228299
A linear model can help us answer questions about association between response and predictors, predict sales in future, linearity of relation, and interaction between predictors.
\[ Y \approx \beta_0 + \beta_1X \]
\[\beta_0\hspace{1mm} is\hspace{1mm}population\hspace{1mm}intercept\]
\[\beta_1\hspace{1mm} is\hspace{1mm}population\hspace{1mm}slope\] Our estimates are represented as :
\[ \hat\beta_0\] \[\hat\beta_1\]
The Idea is to, essentially, draw a line through the points such that distance of every point from line is as small a possible.
One way to get estimates of population coefficients or parameters is minimizing least squares.
\[sales \approx \beta_0 + \beta_1*TV\]
\[\hat y_i = \hat\beta_0 + \hat\beta_1x_i\] \[e_i = y_i - \hat y_i\]
\[RSS = e_1^2 + e_2^2....+e_n^2\]
Least square coefficient estimates
\[ \hat\beta_1 = \frac{\sum_i^n(x_i - \bar x)(y_i - \bar y)}{\sum_i^n(x_i - \bar x)^2} \]
\[ \hat\beta_0 = \bar y - \hat\beta_1\bar x \]
Call:
lm(formula = sales ~ TV, data = advertisement)
Coefficients:
(Intercept) TV
7.03259 0.04754
“For every`additional $1000 spent on TV advertisement budget, there is additional sale of ~47.5 units”
Use the data Auto from the {ISRL2} Fit this model.
\[horsepower = \beta_0 + \beta_1*weight + \epsilon\]
find coeff estimates and residuals: \[\hat\beta_0\] and \[\hat\beta_1\]
\[Compute\hspace{1mm} standard\hspace{1mm}error\hspace{1mm} of\hspace{1mm} \hat\beta_0\hspace{1mm} and\hspace{1mm} \hat\beta_1\]
something like this:
\[Var(\hat\mu) = SE(\hat\mu) = \frac{\sigma^2}{n}\]
but in reality
\[SE(\hat\beta_0)^2 = \sigma^2[\frac{1}{n}+\frac{\bar x^2}{\sum_i^n(x_i - \bar x)^2}]\hspace{2cm}SE(\hat\beta_1)^2 = \frac{\sigma^2}{\sum_i^n(x_i - \bar x)^2}\]
What is sigma here ?
\[what\hspace{1mm} happens\hspace{1mm} when\hspace{1mm} x_i\hspace{1mm} are\hspace{1mm} spread\hspace{1mm} out\hspace{1mm} ?\]
We can use SE to to hypothesis testing. t-statistic is used to do this in practise
\[t = \frac{\hat\beta_1 - 0}{SE(\hat\beta_1)}\]
Call:
lm(formula = sales ~ TV, data = advertisement)
Residuals:
Min 1Q Median 3Q Max
-8.3860 -1.9545 -0.1913 2.0671 7.2124
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.032594 0.457843 15.36 <2e-16 ***
TV 0.047537 0.002691 17.67 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.259 on 198 degrees of freedom
Multiple R-squared: 0.6119, Adjusted R-squared: 0.6099
F-statistic: 312.1 on 1 and 198 DF, p-value: < 2.2e-16
RSE and R^2
[1] 3.258656
[1] 0.2323877
[1] 0.6118751
Residual Standard Error is the standard deviation of \(\epsilon\). Essentially \(\sqrt\frac{RSS}{n-2}\)
This is a measure of lack of fit.
It is in terms of Y, so scale is involved(Beware).
It is the proportion of variance in the response that is explained by the model.
TSS is total sum of squares \(\sum_i^n(y_i - \bar y)\)
\[\frac{TSS-RSS}{TSS}\]
\(R^2\) is between 0 and 1. Therefore easier to interpret.
# A tibble: 4 × 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 2.94 0.312 9.42 1.27e-17
2 TV 0.0458 0.00139 32.8 1.51e-81
3 radio 0.189 0.00861 21.9 1.51e-54
4 newspaper -0.00104 0.00587 -0.177 8.60e- 1
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.897 0.896 1.69 570. 1.58e-96 3 -386. 782. 799.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
# A tibble: 200 × 10
sales TV radio newspaper .fitted .resid .hat .sigma .cooksd .std.resid
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 22.1 230. 37.8 69.2 20.5 1.58 0.0252 1.69 5.80e-3 0.947
2 10.4 44.5 39.3 45.1 12.3 -1.94 0.0194 1.68 6.67e-3 -1.16
3 9.3 17.2 45.9 69.3 12.3 -3.01 0.0392 1.68 3.38e-2 -1.82
4 18.5 152. 41.3 58.5 17.6 0.902 0.0166 1.69 1.23e-3 0.540
5 12.9 181. 10.8 58.4 13.2 -0.289 0.0235 1.69 1.81e-4 -0.173
6 7.2 8.7 48.9 75 12.5 -5.28 0.0475 1.64 1.28e-1 -3.21
7 11.8 57.5 32.8 23.5 11.7 0.0702 0.0144 1.69 6.45e-6 0.0420
8 13.2 120. 19.6 11.6 12.1 1.08 0.00918 1.69 9.55e-4 0.642
9 4.8 8.6 2.1 1 3.73 1.07 0.0307 1.69 3.31e-3 0.646
10 10.6 200. 2.6 21.2 12.6 -1.95 0.0171 1.68 5.95e-3 -1.17
# ℹ 190 more rows
lm(sales ~ newspaper, data = advertisement)? Why is this so different?\[ F = \frac{(TSS-RSS)/p}{RSS/(n-p-1)} \] In a multiple linear regression we need to ensure that all coefficients for \(p\) predictors are not equal to zero. Meaning that at least one of those is non-zero. F-statistic helps us ensure that.
That means if \(H_0\) is true, F will be near 1 and if not F will be far larger than 1.
Well if one or more predictors have non-zero coef and are associated with response, Which are those?
| Income | Limit | Rating | Cards | Age | Education | Own | Student | Married | Region | Balance |
|---|---|---|---|---|---|---|---|---|---|---|
| 14.891 | 3606 | 283 | 2 | 34 | 11 | No | No | Yes | South | 333 |
| 106.025 | 6645 | 483 | 3 | 82 | 15 | Yes | Yes | Yes | West | 903 |
| 104.593 | 7075 | 514 | 4 | 71 | 11 | No | No | No | West | 580 |
| 148.924 | 9504 | 681 | 3 | 36 | 11 | Yes | No | No | West | 964 |
| 55.882 | 4897 | 357 | 2 | 68 | 16 | No | No | Yes | South | 331 |
| 80.180 | 8047 | 569 | 4 | 77 | 10 | No | No | No | South | 1151 |
| 20.996 | 3388 | 259 | 2 | 37 | 12 | Yes | No | No | East | 203 |
| 71.408 | 7114 | 512 | 2 | 87 | 9 | No | No | No | West | 872 |
| 15.125 | 3300 | 266 | 5 | 66 | 13 | Yes | No | No | South | 279 |
| 71.061 | 6819 | 491 | 3 | 41 | 19 | Yes | Yes | Yes | East | 1350 |
| 63.095 | 8117 | 589 | 4 | 30 | 14 | No | No | Yes | South | 1407 |
| 15.045 | 1311 | 138 | 3 | 64 | 16 | No | No | No | South | 0 |
| 80.616 | 5308 | 394 | 1 | 57 | 7 | Yes | No | Yes | West | 204 |
| 43.682 | 6922 | 511 | 1 | 49 | 9 | No | No | Yes | South | 1081 |
| 19.144 | 3291 | 269 | 2 | 75 | 13 | Yes | No | No | East | 148 |
| 20.089 | 2525 | 200 | 3 | 57 | 15 | Yes | No | Yes | East | 0 |
| 53.598 | 3714 | 286 | 3 | 73 | 17 | Yes | No | Yes | East | 0 |
| 36.496 | 4378 | 339 | 3 | 69 | 15 | Yes | No | Yes | West | 368 |
| 49.570 | 6384 | 448 | 1 | 28 | 9 | Yes | No | Yes | West | 891 |
| 42.079 | 6626 | 479 | 2 | 44 | 9 | No | No | No | West | 1048 |
| 17.700 | 2860 | 235 | 4 | 63 | 16 | Yes | No | No | West | 89 |
| 37.348 | 6378 | 458 | 1 | 72 | 17 | Yes | No | No | South | 968 |
| 20.103 | 2631 | 213 | 3 | 61 | 10 | No | No | Yes | East | 0 |
| 64.027 | 5179 | 398 | 5 | 48 | 8 | No | No | Yes | East | 411 |
| 10.742 | 1757 | 156 | 3 | 57 | 15 | Yes | No | No | South | 0 |
| 14.090 | 4323 | 326 | 5 | 25 | 16 | Yes | No | Yes | East | 671 |
| 42.471 | 3625 | 289 | 6 | 44 | 12 | Yes | Yes | No | South | 654 |
| 32.793 | 4534 | 333 | 2 | 44 | 16 | No | No | No | East | 467 |
| 186.634 | 13414 | 949 | 2 | 41 | 14 | Yes | No | Yes | East | 1809 |
| 26.813 | 5611 | 411 | 4 | 55 | 16 | Yes | No | No | South | 915 |
| 34.142 | 5666 | 413 | 4 | 47 | 5 | Yes | No | Yes | South | 863 |
| 28.941 | 2733 | 210 | 5 | 43 | 16 | No | No | Yes | West | 0 |
| 134.181 | 7838 | 563 | 2 | 48 | 13 | Yes | No | No | South | 526 |
| 31.367 | 1829 | 162 | 4 | 30 | 10 | No | No | Yes | South | 0 |
| 20.150 | 2646 | 199 | 2 | 25 | 14 | Yes | No | Yes | West | 0 |
| 23.350 | 2558 | 220 | 3 | 49 | 12 | Yes | Yes | No | South | 419 |
| 62.413 | 6457 | 455 | 2 | 71 | 11 | Yes | No | Yes | South | 762 |
| 30.007 | 6481 | 462 | 2 | 69 | 9 | Yes | No | Yes | South | 1093 |
| 11.795 | 3899 | 300 | 4 | 25 | 10 | Yes | No | No | South | 531 |
| 13.647 | 3461 | 264 | 4 | 47 | 14 | No | No | Yes | South | 344 |
| 34.950 | 3327 | 253 | 3 | 54 | 14 | Yes | No | No | East | 50 |
| 113.659 | 7659 | 538 | 2 | 66 | 15 | No | Yes | Yes | East | 1155 |
| 44.158 | 4763 | 351 | 2 | 66 | 13 | Yes | No | Yes | West | 385 |
| 36.929 | 6257 | 445 | 1 | 24 | 14 | Yes | No | Yes | West | 976 |
| 31.861 | 6375 | 469 | 3 | 25 | 16 | Yes | No | Yes | South | 1120 |
| 77.380 | 7569 | 564 | 3 | 50 | 12 | Yes | No | Yes | South | 997 |
| 19.531 | 5043 | 376 | 2 | 64 | 16 | Yes | Yes | Yes | West | 1241 |
| 44.646 | 4431 | 320 | 2 | 49 | 15 | No | Yes | Yes | South | 797 |
| 44.522 | 2252 | 205 | 6 | 72 | 15 | No | No | Yes | West | 0 |
| 43.479 | 4569 | 354 | 4 | 49 | 13 | No | Yes | Yes | East | 902 |
| 36.362 | 5183 | 376 | 3 | 49 | 15 | No | No | Yes | East | 654 |
| 39.705 | 3969 | 301 | 2 | 27 | 20 | No | No | Yes | East | 211 |
| 44.205 | 5441 | 394 | 1 | 32 | 12 | No | No | Yes | South | 607 |
| 16.304 | 5466 | 413 | 4 | 66 | 10 | No | No | Yes | West | 957 |
| 15.333 | 1499 | 138 | 2 | 47 | 9 | Yes | No | Yes | West | 0 |
| 32.916 | 1786 | 154 | 2 | 60 | 8 | Yes | No | Yes | West | 0 |
| 57.100 | 4742 | 372 | 7 | 79 | 18 | Yes | No | Yes | West | 379 |
| 76.273 | 4779 | 367 | 4 | 65 | 14 | Yes | No | Yes | South | 133 |
| 10.354 | 3480 | 281 | 2 | 70 | 17 | No | No | Yes | South | 333 |
| 51.872 | 5294 | 390 | 4 | 81 | 17 | Yes | No | No | South | 531 |
| 35.510 | 5198 | 364 | 2 | 35 | 20 | Yes | No | No | West | 631 |
| 21.238 | 3089 | 254 | 3 | 59 | 10 | Yes | No | No | South | 108 |
| 30.682 | 1671 | 160 | 2 | 77 | 7 | Yes | No | No | South | 0 |
| 14.132 | 2998 | 251 | 4 | 75 | 17 | No | No | No | South | 133 |
| 32.164 | 2937 | 223 | 2 | 79 | 15 | Yes | No | Yes | East | 0 |
| 12.000 | 4160 | 320 | 4 | 28 | 14 | Yes | No | Yes | South | 602 |
| 113.829 | 9704 | 694 | 4 | 38 | 13 | Yes | No | Yes | West | 1388 |
| 11.187 | 5099 | 380 | 4 | 69 | 16 | Yes | No | No | East | 889 |
| 27.847 | 5619 | 418 | 2 | 78 | 15 | Yes | No | Yes | South | 822 |
| 49.502 | 6819 | 505 | 4 | 55 | 14 | No | No | Yes | South | 1084 |
| 24.889 | 3954 | 318 | 4 | 75 | 12 | No | No | Yes | South | 357 |
| 58.781 | 7402 | 538 | 2 | 81 | 12 | Yes | No | Yes | West | 1103 |
| 22.939 | 4923 | 355 | 1 | 47 | 18 | Yes | No | Yes | West | 663 |
| 23.989 | 4523 | 338 | 4 | 31 | 15 | No | No | No | South | 601 |
| 16.103 | 5390 | 418 | 4 | 45 | 10 | Yes | No | Yes | South | 945 |
| 33.017 | 3180 | 224 | 2 | 28 | 16 | No | No | Yes | East | 29 |
| 30.622 | 3293 | 251 | 1 | 68 | 16 | No | Yes | No | South | 532 |
| 20.936 | 3254 | 253 | 1 | 30 | 15 | Yes | No | No | West | 145 |
| 110.968 | 6662 | 468 | 3 | 45 | 11 | Yes | No | Yes | South | 391 |
| 15.354 | 2101 | 171 | 2 | 65 | 14 | No | No | No | West | 0 |
| 27.369 | 3449 | 288 | 3 | 40 | 9 | Yes | No | Yes | South | 162 |
| 53.480 | 4263 | 317 | 1 | 83 | 15 | No | No | No | South | 99 |
| 23.672 | 4433 | 344 | 3 | 63 | 11 | No | No | No | South | 503 |
| 19.225 | 1433 | 122 | 3 | 38 | 14 | Yes | No | No | South | 0 |
| 43.540 | 2906 | 232 | 4 | 69 | 11 | No | No | No | South | 0 |
| 152.298 | 12066 | 828 | 4 | 41 | 12 | Yes | No | Yes | West | 1779 |
| 55.367 | 6340 | 448 | 1 | 33 | 15 | No | No | Yes | South | 815 |
| 11.741 | 2271 | 182 | 4 | 59 | 12 | Yes | No | No | West | 0 |
| 15.560 | 4307 | 352 | 4 | 57 | 8 | No | No | Yes | East | 579 |
| 59.530 | 7518 | 543 | 3 | 52 | 9 | Yes | No | No | East | 1176 |
| 20.191 | 5767 | 431 | 4 | 42 | 16 | No | No | Yes | East | 1023 |
| 48.498 | 6040 | 456 | 3 | 47 | 16 | No | No | Yes | South | 812 |
| 30.733 | 2832 | 249 | 4 | 51 | 13 | No | No | No | South | 0 |
| 16.479 | 5435 | 388 | 2 | 26 | 16 | No | No | No | East | 937 |
| 38.009 | 3075 | 245 | 3 | 45 | 15 | Yes | No | No | East | 0 |
| 14.084 | 855 | 120 | 5 | 46 | 17 | Yes | No | Yes | East | 0 |
| 14.312 | 5382 | 367 | 1 | 59 | 17 | No | Yes | No | West | 1380 |
| 26.067 | 3388 | 266 | 4 | 74 | 17 | Yes | No | Yes | East | 155 |
| 36.295 | 2963 | 241 | 2 | 68 | 14 | Yes | Yes | No | East | 375 |
| 83.851 | 8494 | 607 | 5 | 47 | 18 | No | No | No | South | 1311 |
| 21.153 | 3736 | 256 | 1 | 41 | 11 | No | No | No | South | 298 |
| 17.976 | 2433 | 190 | 3 | 70 | 16 | Yes | Yes | No | South | 431 |
| 68.713 | 7582 | 531 | 2 | 56 | 16 | No | Yes | No | South | 1587 |
| 146.183 | 9540 | 682 | 6 | 66 | 15 | No | No | No | South | 1050 |
| 15.846 | 4768 | 365 | 4 | 53 | 12 | Yes | No | No | South | 745 |
| 12.031 | 3182 | 259 | 2 | 58 | 18 | Yes | No | Yes | South | 210 |
| 16.819 | 1337 | 115 | 2 | 74 | 15 | No | No | Yes | West | 0 |
| 39.110 | 3189 | 263 | 3 | 72 | 12 | No | No | No | West | 0 |
| 107.986 | 6033 | 449 | 4 | 64 | 14 | No | No | Yes | South | 227 |
| 13.561 | 3261 | 279 | 5 | 37 | 19 | No | No | Yes | West | 297 |
| 34.537 | 3271 | 250 | 3 | 57 | 17 | Yes | No | Yes | West | 47 |
| 28.575 | 2959 | 231 | 2 | 60 | 11 | Yes | No | No | East | 0 |
| 46.007 | 6637 | 491 | 4 | 42 | 14 | No | No | Yes | South | 1046 |
| 69.251 | 6386 | 474 | 4 | 30 | 12 | Yes | No | Yes | West | 768 |
| 16.482 | 3326 | 268 | 4 | 41 | 15 | No | No | No | South | 271 |
| 40.442 | 4828 | 369 | 5 | 81 | 8 | Yes | No | No | East | 510 |
| 35.177 | 2117 | 186 | 3 | 62 | 16 | Yes | No | No | South | 0 |
| 91.362 | 9113 | 626 | 1 | 47 | 17 | No | No | Yes | West | 1341 |
| 27.039 | 2161 | 173 | 3 | 40 | 17 | Yes | No | No | South | 0 |
| 23.012 | 1410 | 137 | 3 | 81 | 16 | No | No | No | South | 0 |
| 27.241 | 1402 | 128 | 2 | 67 | 15 | Yes | No | Yes | West | 0 |
| 148.080 | 8157 | 599 | 2 | 83 | 13 | No | No | Yes | South | 454 |
| 62.602 | 7056 | 481 | 1 | 84 | 11 | Yes | No | No | South | 904 |
| 11.808 | 1300 | 117 | 3 | 77 | 14 | Yes | No | No | East | 0 |
| 29.564 | 2529 | 192 | 1 | 30 | 12 | Yes | No | Yes | South | 0 |
| 27.578 | 2531 | 195 | 1 | 34 | 15 | Yes | No | Yes | South | 0 |
| 26.427 | 5533 | 433 | 5 | 50 | 15 | Yes | Yes | Yes | West | 1404 |
| 57.202 | 3411 | 259 | 3 | 72 | 11 | Yes | No | No | South | 0 |
| 123.299 | 8376 | 610 | 2 | 89 | 17 | No | Yes | No | East | 1259 |
| 18.145 | 3461 | 279 | 3 | 56 | 15 | No | No | Yes | East | 255 |
| 23.793 | 3821 | 281 | 4 | 56 | 12 | Yes | Yes | Yes | East | 868 |
| 10.726 | 1568 | 162 | 5 | 46 | 19 | No | No | Yes | West | 0 |
| 23.283 | 5443 | 407 | 4 | 49 | 13 | No | No | Yes | East | 912 |
| 21.455 | 5829 | 427 | 4 | 80 | 12 | Yes | No | Yes | East | 1018 |
| 34.664 | 5835 | 452 | 3 | 77 | 15 | Yes | No | Yes | East | 835 |
| 44.473 | 3500 | 257 | 3 | 81 | 16 | Yes | No | No | East | 8 |
| 54.663 | 4116 | 314 | 2 | 70 | 8 | Yes | No | No | East | 75 |
| 36.355 | 3613 | 278 | 4 | 35 | 9 | No | No | Yes | West | 187 |
| 21.374 | 2073 | 175 | 2 | 74 | 11 | Yes | No | Yes | South | 0 |
| 107.841 | 10384 | 728 | 3 | 87 | 7 | No | No | No | East | 1597 |
| 39.831 | 6045 | 459 | 3 | 32 | 12 | Yes | Yes | Yes | East | 1425 |
| 91.876 | 6754 | 483 | 2 | 33 | 10 | No | No | Yes | South | 605 |
| 103.893 | 7416 | 549 | 3 | 84 | 17 | No | No | No | West | 669 |
| 19.636 | 4896 | 387 | 3 | 64 | 10 | Yes | No | No | East | 710 |
| 17.392 | 2748 | 228 | 3 | 32 | 14 | No | No | Yes | South | 68 |
| 19.529 | 4673 | 341 | 2 | 51 | 14 | No | No | No | West | 642 |
| 17.055 | 5110 | 371 | 3 | 55 | 15 | Yes | No | Yes | South | 805 |
| 23.857 | 1501 | 150 | 3 | 56 | 16 | No | No | Yes | South | 0 |
| 15.184 | 2420 | 192 | 2 | 69 | 11 | Yes | No | Yes | South | 0 |
| 13.444 | 886 | 121 | 5 | 44 | 10 | No | No | Yes | West | 0 |
| 63.931 | 5728 | 435 | 3 | 28 | 14 | Yes | No | Yes | East | 581 |
| 35.864 | 4831 | 353 | 3 | 66 | 13 | Yes | No | Yes | South | 534 |
| 41.419 | 2120 | 184 | 4 | 24 | 11 | Yes | Yes | No | South | 156 |
| 92.112 | 4612 | 344 | 3 | 32 | 17 | No | No | No | South | 0 |
| 55.056 | 3155 | 235 | 2 | 31 | 16 | No | No | Yes | East | 0 |
| 19.537 | 1362 | 143 | 4 | 34 | 9 | Yes | No | Yes | West | 0 |
| 31.811 | 4284 | 338 | 5 | 75 | 13 | Yes | No | Yes | South | 429 |
| 56.256 | 5521 | 406 | 2 | 72 | 16 | Yes | Yes | Yes | South | 1020 |
| 42.357 | 5550 | 406 | 2 | 83 | 12 | Yes | No | Yes | West | 653 |
| 53.319 | 3000 | 235 | 3 | 53 | 13 | No | No | No | West | 0 |
| 12.238 | 4865 | 381 | 5 | 67 | 11 | Yes | No | No | South | 836 |
| 31.353 | 1705 | 160 | 3 | 81 | 14 | No | No | Yes | South | 0 |
| 63.809 | 7530 | 515 | 1 | 56 | 12 | No | No | Yes | South | 1086 |
| 13.676 | 2330 | 203 | 5 | 80 | 16 | Yes | No | No | East | 0 |
| 76.782 | 5977 | 429 | 4 | 44 | 12 | No | No | Yes | West | 548 |
| 25.383 | 4527 | 367 | 4 | 46 | 11 | No | No | Yes | South | 570 |
| 35.691 | 2880 | 214 | 2 | 35 | 15 | No | No | No | East | 0 |
| 29.403 | 2327 | 178 | 1 | 37 | 14 | Yes | No | Yes | South | 0 |
| 27.470 | 2820 | 219 | 1 | 32 | 11 | Yes | No | Yes | West | 0 |
| 27.330 | 6179 | 459 | 4 | 36 | 12 | Yes | No | Yes | South | 1099 |
| 34.772 | 2021 | 167 | 3 | 57 | 9 | No | No | No | West | 0 |
| 36.934 | 4270 | 299 | 1 | 63 | 9 | Yes | No | Yes | South | 283 |
| 76.348 | 4697 | 344 | 4 | 60 | 18 | No | No | No | West | 108 |
| 14.887 | 4745 | 339 | 3 | 58 | 12 | No | No | Yes | East | 724 |
| 121.834 | 10673 | 750 | 3 | 54 | 16 | No | No | No | East | 1573 |
| 30.132 | 2168 | 206 | 3 | 52 | 17 | No | No | No | South | 0 |
| 24.050 | 2607 | 221 | 4 | 32 | 18 | No | No | Yes | South | 0 |
| 22.379 | 3965 | 292 | 2 | 34 | 14 | Yes | No | Yes | West | 384 |
| 28.316 | 4391 | 316 | 2 | 29 | 10 | Yes | No | No | South | 453 |
| 58.026 | 7499 | 560 | 5 | 67 | 11 | Yes | No | No | South | 1237 |
| 10.635 | 3584 | 294 | 5 | 69 | 16 | No | No | Yes | West | 423 |
| 46.102 | 5180 | 382 | 3 | 81 | 12 | No | No | Yes | East | 516 |
| 58.929 | 6420 | 459 | 2 | 66 | 9 | Yes | No | Yes | East | 789 |
| 80.861 | 4090 | 335 | 3 | 29 | 15 | Yes | No | Yes | West | 0 |
| 158.889 | 11589 | 805 | 1 | 62 | 17 | Yes | No | Yes | South | 1448 |
| 30.420 | 4442 | 316 | 1 | 30 | 14 | Yes | No | No | East | 450 |
| 36.472 | 3806 | 309 | 2 | 52 | 13 | No | No | No | East | 188 |
| 23.365 | 2179 | 167 | 2 | 75 | 15 | No | No | No | West | 0 |
| 83.869 | 7667 | 554 | 2 | 83 | 11 | No | No | No | East | 930 |
| 58.351 | 4411 | 326 | 2 | 85 | 16 | Yes | No | Yes | South | 126 |
| 55.187 | 5352 | 385 | 4 | 50 | 17 | Yes | No | Yes | South | 538 |
| 124.290 | 9560 | 701 | 3 | 52 | 17 | Yes | Yes | No | West | 1687 |
| 28.508 | 3933 | 287 | 4 | 56 | 14 | No | No | Yes | West | 336 |
| 130.209 | 10088 | 730 | 7 | 39 | 19 | Yes | No | Yes | South | 1426 |
| 30.406 | 2120 | 181 | 2 | 79 | 14 | No | No | Yes | East | 0 |
| 23.883 | 5384 | 398 | 2 | 73 | 16 | Yes | No | Yes | East | 802 |
| 93.039 | 7398 | 517 | 1 | 67 | 12 | No | No | Yes | East | 749 |
| 50.699 | 3977 | 304 | 2 | 84 | 17 | Yes | No | No | East | 69 |
| 27.349 | 2000 | 169 | 4 | 51 | 16 | Yes | No | Yes | East | 0 |
| 10.403 | 4159 | 310 | 3 | 43 | 7 | No | No | Yes | West | 571 |
| 23.949 | 5343 | 383 | 2 | 40 | 18 | No | No | Yes | East | 829 |
| 73.914 | 7333 | 529 | 6 | 67 | 15 | Yes | No | Yes | South | 1048 |
| 21.038 | 1448 | 145 | 2 | 58 | 13 | Yes | No | Yes | South | 0 |
| 68.206 | 6784 | 499 | 5 | 40 | 16 | Yes | Yes | No | East | 1411 |
| 57.337 | 5310 | 392 | 2 | 45 | 7 | Yes | No | No | South | 456 |
| 10.793 | 3878 | 321 | 8 | 29 | 13 | No | No | No | South | 638 |
| 23.450 | 2450 | 180 | 2 | 78 | 13 | No | No | No | South | 0 |
| 10.842 | 4391 | 358 | 5 | 37 | 10 | Yes | Yes | Yes | South | 1216 |
| 51.345 | 4327 | 320 | 3 | 46 | 15 | No | No | No | East | 230 |
| 151.947 | 9156 | 642 | 2 | 91 | 11 | Yes | No | Yes | East | 732 |
| 24.543 | 3206 | 243 | 2 | 62 | 12 | Yes | No | Yes | South | 95 |
| 29.567 | 5309 | 397 | 3 | 25 | 15 | No | No | No | South | 799 |
| 39.145 | 4351 | 323 | 2 | 66 | 13 | No | No | Yes | South | 308 |
| 39.422 | 5245 | 383 | 2 | 44 | 19 | No | No | No | East | 637 |
| 34.909 | 5289 | 410 | 2 | 62 | 16 | Yes | No | Yes | South | 681 |
| 41.025 | 4229 | 337 | 3 | 79 | 19 | Yes | No | Yes | South | 246 |
| 15.476 | 2762 | 215 | 3 | 60 | 18 | No | No | No | West | 52 |
| 12.456 | 5395 | 392 | 3 | 65 | 14 | No | No | Yes | South | 955 |
| 10.627 | 1647 | 149 | 2 | 71 | 10 | Yes | Yes | Yes | West | 195 |
| 38.954 | 5222 | 370 | 4 | 76 | 13 | Yes | No | No | South | 653 |
| 44.847 | 5765 | 437 | 3 | 53 | 13 | Yes | Yes | No | West | 1246 |
| 98.515 | 8760 | 633 | 5 | 78 | 11 | Yes | No | No | East | 1230 |
| 33.437 | 6207 | 451 | 4 | 44 | 9 | No | Yes | No | South | 1549 |
| 27.512 | 4613 | 344 | 5 | 72 | 17 | No | No | Yes | West | 573 |
| 121.709 | 7818 | 584 | 4 | 50 | 6 | No | No | Yes | South | 701 |
| 15.079 | 5673 | 411 | 4 | 28 | 15 | Yes | No | Yes | West | 1075 |
| 59.879 | 6906 | 527 | 6 | 78 | 15 | Yes | No | No | South | 1032 |
| 66.989 | 5614 | 430 | 3 | 47 | 14 | Yes | No | Yes | South | 482 |
| 69.165 | 4668 | 341 | 2 | 34 | 11 | Yes | No | No | East | 156 |
| 69.943 | 7555 | 547 | 3 | 76 | 9 | No | No | Yes | West | 1058 |
| 33.214 | 5137 | 387 | 3 | 59 | 9 | No | No | No | East | 661 |
| 25.124 | 4776 | 378 | 4 | 29 | 12 | No | No | Yes | South | 657 |
| 15.741 | 4788 | 360 | 1 | 39 | 14 | No | No | Yes | West | 689 |
| 11.603 | 2278 | 187 | 3 | 71 | 11 | No | No | Yes | South | 0 |
| 69.656 | 8244 | 579 | 3 | 41 | 14 | No | No | Yes | East | 1329 |
| 10.503 | 2923 | 232 | 3 | 25 | 18 | Yes | No | Yes | East | 191 |
| 42.529 | 4986 | 369 | 2 | 37 | 11 | No | No | Yes | West | 489 |
| 60.579 | 5149 | 388 | 5 | 38 | 15 | No | No | Yes | West | 443 |
| 26.532 | 2910 | 236 | 6 | 58 | 19 | Yes | No | Yes | South | 52 |
| 27.952 | 3557 | 263 | 1 | 35 | 13 | Yes | No | Yes | West | 163 |
| 29.705 | 3351 | 262 | 5 | 71 | 14 | Yes | No | Yes | West | 148 |
| 15.602 | 906 | 103 | 2 | 36 | 11 | No | No | Yes | East | 0 |
| 20.918 | 1233 | 128 | 3 | 47 | 18 | Yes | Yes | Yes | West | 16 |
| 58.165 | 6617 | 460 | 1 | 56 | 12 | Yes | No | Yes | South | 856 |
| 22.561 | 1787 | 147 | 4 | 66 | 15 | Yes | No | No | South | 0 |
| 34.509 | 2001 | 189 | 5 | 80 | 18 | Yes | No | Yes | East | 0 |
| 19.588 | 3211 | 265 | 4 | 59 | 14 | Yes | No | No | West | 199 |
| 36.364 | 2220 | 188 | 3 | 50 | 19 | No | No | No | South | 0 |
| 15.717 | 905 | 93 | 1 | 38 | 16 | No | Yes | Yes | South | 0 |
| 22.574 | 1551 | 134 | 3 | 43 | 13 | Yes | Yes | Yes | South | 98 |
| 10.363 | 2430 | 191 | 2 | 47 | 18 | Yes | No | Yes | West | 0 |
| 28.474 | 3202 | 267 | 5 | 66 | 12 | No | No | Yes | South | 132 |
| 72.945 | 8603 | 621 | 3 | 64 | 8 | Yes | No | No | South | 1355 |
| 85.425 | 5182 | 402 | 6 | 60 | 12 | No | No | Yes | East | 218 |
| 36.508 | 6386 | 469 | 4 | 79 | 6 | Yes | No | Yes | South | 1048 |
| 58.063 | 4221 | 304 | 3 | 50 | 8 | No | No | No | East | 118 |
| 25.936 | 1774 | 135 | 2 | 71 | 14 | Yes | No | No | West | 0 |
| 15.629 | 2493 | 186 | 1 | 60 | 14 | No | No | Yes | West | 0 |
| 41.400 | 2561 | 215 | 2 | 36 | 14 | No | No | Yes | South | 0 |
| 33.657 | 6196 | 450 | 6 | 55 | 9 | Yes | No | No | South | 1092 |
| 67.937 | 5184 | 383 | 4 | 63 | 12 | No | No | Yes | West | 345 |
| 180.379 | 9310 | 665 | 3 | 67 | 8 | Yes | Yes | Yes | West | 1050 |
| 10.588 | 4049 | 296 | 1 | 66 | 13 | Yes | No | Yes | South | 465 |
| 29.725 | 3536 | 270 | 2 | 52 | 15 | Yes | No | No | East | 133 |
| 27.999 | 5107 | 380 | 1 | 55 | 10 | No | No | Yes | South | 651 |
| 40.885 | 5013 | 379 | 3 | 46 | 13 | Yes | No | Yes | East | 549 |
| 88.830 | 4952 | 360 | 4 | 86 | 16 | Yes | No | Yes | South | 15 |
| 29.638 | 5833 | 433 | 3 | 29 | 15 | Yes | No | Yes | West | 942 |
| 25.988 | 1349 | 142 | 4 | 82 | 12 | No | No | No | South | 0 |
| 39.055 | 5565 | 410 | 4 | 48 | 18 | Yes | No | Yes | South | 772 |
| 15.866 | 3085 | 217 | 1 | 39 | 13 | No | No | No | South | 136 |
| 44.978 | 4866 | 347 | 1 | 30 | 10 | Yes | No | No | South | 436 |
| 30.413 | 3690 | 299 | 2 | 25 | 15 | Yes | Yes | No | West | 728 |
| 16.751 | 4706 | 353 | 6 | 48 | 14 | No | Yes | No | West | 1255 |
| 30.550 | 5869 | 439 | 5 | 81 | 9 | Yes | No | No | East | 967 |
| 163.329 | 8732 | 636 | 3 | 50 | 14 | No | No | Yes | South | 529 |
| 23.106 | 3476 | 257 | 2 | 50 | 15 | Yes | No | No | South | 209 |
| 41.532 | 5000 | 353 | 2 | 50 | 12 | No | No | Yes | South | 531 |
| 128.040 | 6982 | 518 | 2 | 78 | 11 | Yes | No | Yes | South | 250 |
| 54.319 | 3063 | 248 | 3 | 59 | 8 | Yes | Yes | No | South | 269 |
| 53.401 | 5319 | 377 | 3 | 35 | 12 | Yes | No | No | East | 541 |
| 36.142 | 1852 | 183 | 3 | 33 | 13 | Yes | No | No | East | 0 |
| 63.534 | 8100 | 581 | 2 | 50 | 17 | Yes | No | Yes | South | 1298 |
| 49.927 | 6396 | 485 | 3 | 75 | 17 | Yes | No | Yes | South | 890 |
| 14.711 | 2047 | 167 | 2 | 67 | 6 | No | No | Yes | South | 0 |
| 18.967 | 1626 | 156 | 2 | 41 | 11 | Yes | No | Yes | West | 0 |
| 18.036 | 1552 | 142 | 2 | 48 | 15 | Yes | No | No | South | 0 |
| 60.449 | 3098 | 272 | 4 | 69 | 8 | No | No | Yes | South | 0 |
| 16.711 | 5274 | 387 | 3 | 42 | 16 | Yes | No | Yes | West | 863 |
| 10.852 | 3907 | 296 | 2 | 30 | 9 | No | No | No | South | 485 |
| 26.370 | 3235 | 268 | 5 | 78 | 11 | No | No | Yes | West | 159 |
| 24.088 | 3665 | 287 | 4 | 56 | 13 | Yes | No | Yes | South | 309 |
| 51.532 | 5096 | 380 | 2 | 31 | 15 | No | No | Yes | South | 481 |
| 140.672 | 11200 | 817 | 7 | 46 | 9 | No | No | Yes | East | 1677 |
| 42.915 | 2532 | 205 | 4 | 42 | 13 | No | No | Yes | West | 0 |
| 27.272 | 1389 | 149 | 5 | 67 | 10 | Yes | No | Yes | South | 0 |
| 65.896 | 5140 | 370 | 1 | 49 | 17 | Yes | No | Yes | South | 293 |
| 55.054 | 4381 | 321 | 3 | 74 | 17 | No | No | Yes | West | 188 |
| 20.791 | 2672 | 204 | 1 | 70 | 18 | Yes | No | No | East | 0 |
| 24.919 | 5051 | 372 | 3 | 76 | 11 | Yes | No | Yes | East | 711 |
| 21.786 | 4632 | 355 | 1 | 50 | 17 | No | No | Yes | South | 580 |
| 31.335 | 3526 | 289 | 3 | 38 | 7 | Yes | No | No | South | 172 |
| 59.855 | 4964 | 365 | 1 | 46 | 13 | Yes | No | Yes | South | 295 |
| 44.061 | 4970 | 352 | 1 | 79 | 11 | No | No | Yes | East | 414 |
| 82.706 | 7506 | 536 | 2 | 64 | 13 | Yes | No | Yes | West | 905 |
| 24.460 | 1924 | 165 | 2 | 50 | 14 | Yes | No | Yes | West | 0 |
| 45.120 | 3762 | 287 | 3 | 80 | 8 | No | No | Yes | South | 70 |
| 75.406 | 3874 | 298 | 3 | 41 | 14 | Yes | No | Yes | West | 0 |
| 14.956 | 4640 | 332 | 2 | 33 | 6 | No | No | No | West | 681 |
| 75.257 | 7010 | 494 | 3 | 34 | 18 | Yes | No | Yes | South | 885 |
| 33.694 | 4891 | 369 | 1 | 52 | 16 | No | Yes | No | East | 1036 |
| 23.375 | 5429 | 396 | 3 | 57 | 15 | Yes | No | Yes | South | 844 |
| 27.825 | 5227 | 386 | 6 | 63 | 11 | No | No | Yes | South | 823 |
| 92.386 | 7685 | 534 | 2 | 75 | 18 | Yes | No | Yes | West | 843 |
| 115.520 | 9272 | 656 | 2 | 69 | 14 | No | No | No | East | 1140 |
| 14.479 | 3907 | 296 | 3 | 43 | 16 | No | No | Yes | South | 463 |
| 52.179 | 7306 | 522 | 2 | 57 | 14 | No | No | No | West | 1142 |
| 68.462 | 4712 | 340 | 2 | 71 | 16 | No | No | Yes | South | 136 |
| 18.951 | 1485 | 129 | 3 | 82 | 13 | Yes | No | No | South | 0 |
| 27.590 | 2586 | 229 | 5 | 54 | 16 | No | No | Yes | East | 0 |
| 16.279 | 1160 | 126 | 3 | 78 | 13 | No | Yes | Yes | East | 5 |
| 25.078 | 3096 | 236 | 2 | 27 | 15 | Yes | No | Yes | South | 81 |
| 27.229 | 3484 | 282 | 6 | 51 | 11 | No | No | No | South | 265 |
| 182.728 | 13913 | 982 | 4 | 98 | 17 | No | No | Yes | South | 1999 |
| 31.029 | 2863 | 223 | 2 | 66 | 17 | No | Yes | Yes | West | 415 |
| 17.765 | 5072 | 364 | 1 | 66 | 12 | Yes | No | Yes | South | 732 |
| 125.480 | 10230 | 721 | 3 | 82 | 16 | No | No | Yes | South | 1361 |
| 49.166 | 6662 | 508 | 3 | 68 | 14 | Yes | No | No | West | 984 |
| 41.192 | 3673 | 297 | 3 | 54 | 16 | Yes | No | Yes | South | 121 |
| 94.193 | 7576 | 527 | 2 | 44 | 16 | Yes | No | Yes | South | 846 |
| 20.405 | 4543 | 329 | 2 | 72 | 17 | No | Yes | No | West | 1054 |
| 12.581 | 3976 | 291 | 2 | 48 | 16 | No | No | Yes | South | 474 |
| 62.328 | 5228 | 377 | 3 | 83 | 15 | No | No | No | South | 380 |
| 21.011 | 3402 | 261 | 2 | 68 | 17 | No | No | Yes | East | 182 |
| 24.230 | 4756 | 351 | 2 | 64 | 15 | Yes | No | Yes | South | 594 |
| 24.314 | 3409 | 270 | 2 | 23 | 7 | Yes | No | Yes | South | 194 |
| 32.856 | 5884 | 438 | 4 | 68 | 13 | No | No | No | South | 926 |
| 12.414 | 855 | 119 | 3 | 32 | 12 | No | No | Yes | East | 0 |
| 41.365 | 5303 | 377 | 1 | 45 | 14 | No | No | No | South | 606 |
| 149.316 | 10278 | 707 | 1 | 80 | 16 | No | No | No | East | 1107 |
| 27.794 | 3807 | 301 | 4 | 35 | 8 | Yes | No | Yes | East | 320 |
| 13.234 | 3922 | 299 | 2 | 77 | 17 | Yes | No | Yes | South | 426 |
| 14.595 | 2955 | 260 | 5 | 37 | 9 | No | No | Yes | East | 204 |
| 10.735 | 3746 | 280 | 2 | 44 | 17 | Yes | No | Yes | South | 410 |
| 48.218 | 5199 | 401 | 7 | 39 | 10 | No | No | Yes | West | 633 |
| 30.012 | 1511 | 137 | 2 | 33 | 17 | No | No | Yes | South | 0 |
| 21.551 | 5380 | 420 | 5 | 51 | 18 | No | No | Yes | West | 907 |
| 160.231 | 10748 | 754 | 2 | 69 | 17 | No | No | No | South | 1192 |
| 13.433 | 1134 | 112 | 3 | 70 | 14 | No | No | Yes | South | 0 |
| 48.577 | 5145 | 389 | 3 | 71 | 13 | Yes | No | Yes | West | 503 |
| 30.002 | 1561 | 155 | 4 | 70 | 13 | Yes | No | Yes | South | 0 |
| 61.620 | 5140 | 374 | 1 | 71 | 9 | No | No | Yes | South | 302 |
| 104.483 | 7140 | 507 | 2 | 41 | 14 | No | No | Yes | East | 583 |
| 41.868 | 4716 | 342 | 2 | 47 | 18 | No | No | No | South | 425 |
| 12.068 | 3873 | 292 | 1 | 44 | 18 | Yes | No | Yes | West | 413 |
| 180.682 | 11966 | 832 | 2 | 58 | 8 | Yes | No | Yes | East | 1405 |
| 34.480 | 6090 | 442 | 3 | 36 | 14 | No | No | No | South | 962 |
| 39.609 | 2539 | 188 | 1 | 40 | 14 | No | No | Yes | West | 0 |
| 30.111 | 4336 | 339 | 1 | 81 | 18 | No | No | Yes | South | 347 |
| 12.335 | 4471 | 344 | 3 | 79 | 12 | No | No | Yes | East | 611 |
| 53.566 | 5891 | 434 | 4 | 82 | 10 | Yes | No | No | South | 712 |
| 53.217 | 4943 | 362 | 2 | 46 | 16 | Yes | No | Yes | West | 382 |
| 26.162 | 5101 | 382 | 3 | 62 | 19 | Yes | No | No | East | 710 |
| 64.173 | 6127 | 433 | 1 | 80 | 10 | No | No | Yes | South | 578 |
| 128.669 | 9824 | 685 | 3 | 67 | 16 | No | No | Yes | West | 1243 |
| 113.772 | 6442 | 489 | 4 | 69 | 15 | No | Yes | Yes | South | 790 |
| 61.069 | 7871 | 564 | 3 | 56 | 14 | No | No | Yes | South | 1264 |
| 23.793 | 3615 | 263 | 2 | 70 | 14 | No | No | No | East | 216 |
| 89.000 | 5759 | 440 | 3 | 37 | 6 | Yes | No | No | South | 345 |
| 71.682 | 8028 | 599 | 3 | 57 | 16 | No | No | Yes | South | 1208 |
| 35.610 | 6135 | 466 | 4 | 40 | 12 | No | No | No | South | 992 |
| 39.116 | 2150 | 173 | 4 | 75 | 15 | No | No | No | South | 0 |
| 19.782 | 3782 | 293 | 2 | 46 | 16 | Yes | Yes | No | South | 840 |
| 55.412 | 5354 | 383 | 2 | 37 | 16 | Yes | Yes | Yes | South | 1003 |
| 29.400 | 4840 | 368 | 3 | 76 | 18 | Yes | No | Yes | South | 588 |
| 20.974 | 5673 | 413 | 5 | 44 | 16 | Yes | No | Yes | South | 1000 |
| 87.625 | 7167 | 515 | 2 | 46 | 10 | Yes | No | No | East | 767 |
| 28.144 | 1567 | 142 | 3 | 51 | 10 | No | No | Yes | South | 0 |
| 19.349 | 4941 | 366 | 1 | 33 | 19 | No | No | Yes | South | 717 |
| 53.308 | 2860 | 214 | 1 | 84 | 10 | No | No | Yes | South | 0 |
| 115.123 | 7760 | 538 | 3 | 83 | 14 | Yes | No | No | East | 661 |
| 101.788 | 8029 | 574 | 2 | 84 | 11 | No | No | Yes | South | 849 |
| 24.824 | 5495 | 409 | 1 | 33 | 9 | No | Yes | No | South | 1352 |
| 14.292 | 3274 | 282 | 9 | 64 | 9 | No | No | Yes | South | 382 |
| 20.088 | 1870 | 180 | 3 | 76 | 16 | No | No | No | East | 0 |
| 26.400 | 5640 | 398 | 3 | 58 | 15 | Yes | No | No | West | 905 |
| 19.253 | 3683 | 287 | 4 | 57 | 10 | No | No | No | East | 371 |
| 16.529 | 1357 | 126 | 3 | 62 | 9 | No | No | No | West | 0 |
| 37.878 | 6827 | 482 | 2 | 80 | 13 | Yes | No | No | South | 1129 |
| 83.948 | 7100 | 503 | 2 | 44 | 18 | No | No | No | South | 806 |
| 135.118 | 10578 | 747 | 3 | 81 | 15 | Yes | No | Yes | West | 1393 |
| 73.327 | 6555 | 472 | 2 | 43 | 15 | Yes | No | No | South | 721 |
| 25.974 | 2308 | 196 | 2 | 24 | 10 | No | No | No | West | 0 |
| 17.316 | 1335 | 138 | 2 | 65 | 13 | No | No | No | East | 0 |
| 49.794 | 5758 | 410 | 4 | 40 | 8 | No | No | No | South | 734 |
| 12.096 | 4100 | 307 | 3 | 32 | 13 | No | No | Yes | South | 560 |
| 13.364 | 3838 | 296 | 5 | 65 | 17 | No | No | No | East | 480 |
| 57.872 | 4171 | 321 | 5 | 67 | 12 | Yes | No | Yes | South | 138 |
| 37.728 | 2525 | 192 | 1 | 44 | 13 | No | No | Yes | South | 0 |
| 18.701 | 5524 | 415 | 5 | 64 | 7 | Yes | No | No | West | 966 |
Use Credit in {ISLR2} data, use balance as response and create a model that you think is good. Try out qualitative variables as well. Begin by looking at all possible variables in data and think which could affect the response.
TRY the {broom} package as well.
We will see some “classic” ways of relaxing these properties.
Instead of \(Y = \beta_0 + \beta_1X_1 + \beta_2X_2\)
We add a interaction term for \(X_1\) and \(X_2\)
\[ \begin{align} \begin{aligned} Y & = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_1X_2\\ & = \beta_0 + X_1(\beta_1 + \beta_3X_2)+\beta_2X_2 \end{aligned} \end{align} \]
What to do when p-values of \(\beta_1\) and \(\beta_2\) are large and the p-value of the interaction term is small? (hierarchical principle)
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 2.92109991 | 0.294489678 | 9.919193 | 4.565557e-19 |
| TV | 0.04575482 | 0.001390356 | 32.908708 | 5.436980e-82 |
| radio | 0.18799423 | 0.008039973 | 23.382446 | 9.776972e-59 |
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.897 0.896 1.68 860. 4.83e-98 2 -386. 780. 794.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 6.750220203 | 0.2478713699 | 27.232755 | 1.541461e-68 |
| TV | 0.019101074 | 0.0015041455 | 12.698953 | 2.363605e-27 |
| radio | 0.028860340 | 0.0089052729 | 3.240815 | 1.400461e-03 |
| TV:radio | 0.001086495 | 0.0000524204 | 20.726564 | 2.757681e-51 |
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.968 0.967 0.944 1963. 6.68e-146 3 -270. 550. 567.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
Use the Credit data in the {ISRL2} library.
How does interation work when the variables of interest are QL * QN or QL * QL
Create a dummy for student
compare these two models
\[Y = \beta_0 + \beta_1income + \beta_2dummy\_student + \beta_3(income*dummy\_student)\] and \[Y = \beta_0 + \beta_1income + \beta_2dummy\_student\]
\(mpg ~ \beta_0 + \beta_1horsepower + \beta_2horsepower^2\)
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 56.900099702 | 1.8004268063 | 31.60367 | 1.740911e-109 |
| horsepower | -0.466189630 | 0.0311246171 | -14.97816 | 2.289429e-40 |
| horsepower_2 | 0.001230536 | 0.0001220759 | 10.08009 | 2.196340e-21 |
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.688 0.686 4.37 428. 5.40e-99 2 -1133. 2274. 2290.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
\(mpg ~ \beta_0 + \beta_1horsepower\)
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 39.9358610 | 0.717498656 | 55.65984 | 1.220362e-187 |
| horsepower | -0.1578447 | 0.006445501 | -24.48914 | 7.031989e-81 |
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.606 0.605 4.91 600. 7.03e-81 1 -1179. 2363. 2375.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
Residual vs Fitted, if there is indication of non-linear relation, try non-linear transformations on you predictors.